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October  2020, 25(10): 3857-3887. doi: 10.3934/dcdsb.2020228

Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media

Yves Capdeboscq 1, and  Shaun Chen Yang Ong 2,

1. 

Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France
2. 

Mathematical Institute, University of Oxford, OX2 6GG, UK

Received  June 2019 Revised  April 2020 Published  July 2020

We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes both positive and negatives values. In this work, we construct a class of inclusions $ Q $ and boundary conditions $ \phi $ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.

 

Erratum: The name of the second author has been corrected from Haun Chen Yang Ong to Shaun Chen Yang Ong. We apologize for any inconvenience this may cause.

Keywords: Conductivity equation, vanishing Jacobian, Rado–Choquet–Kneser theorem, heterogeneous media, Jacobian sign changing structures.
Mathematics Subject Classification: Primary: 35B30, 35B05; Secondary: 35B27, 35R30.
Citation: Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228
References:
[1]

G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018.  Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar

[3]

G. Alessandrini and V. Nesi, Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.  doi: 10.14529/mmp150302.  Google Scholar

[4]

G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242.  Google Scholar

[5]

H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495. doi: 10.24033/asens.2249.  Google Scholar

[6]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. doi: 10.1007/b98245.  Google Scholar

[7]

E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar

[8]

L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.  doi: 10.1007/s00205-010-0302-1.  Google Scholar

[9]

M. Briane and G. W. Milton, Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8.  Google Scholar

[10]

Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178. doi: 10.5802/jep.21.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[12]

S. Friedland, Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.  doi: 10.1080/03081088208817475.  Google Scholar

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[14]

H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93 (2016), 519–545. doi: 10.1112/jlms/jdw003.  Google Scholar

[15]

R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.  doi: 10.1080/17476939608814865.  Google Scholar

[16]

Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar

[17]

A. D. Melas, An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.  doi: 10.1090/S0002-9939-1993-1112497-9.  Google Scholar

[18]

K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968). doi: 10.7146/math.scand.a-10841.  Google Scholar

[19]

J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992). doi: 10.7146/math.scand.a-12375.  Google Scholar

show all references

References:
[1]

G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018.  Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar

[3]

G. Alessandrini and V. Nesi, Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.  doi: 10.14529/mmp150302.  Google Scholar

[4]

G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242.  Google Scholar

[5]

H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495. doi: 10.24033/asens.2249.  Google Scholar

[6]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. doi: 10.1007/b98245.  Google Scholar

[7]

E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar

[8]

L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.  doi: 10.1007/s00205-010-0302-1.  Google Scholar

[9]

M. Briane and G. W. Milton, Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8.  Google Scholar

[10]

Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178. doi: 10.5802/jep.21.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[12]

S. Friedland, Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.  doi: 10.1080/03081088208817475.  Google Scholar

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[14]

H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93 (2016), 519–545. doi: 10.1112/jlms/jdw003.  Google Scholar

[15]

R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.  doi: 10.1080/17476939608814865.  Google Scholar

[16]

Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar

[17]

A. D. Melas, An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.  doi: 10.1090/S0002-9939-1993-1112497-9.  Google Scholar

[18]

K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968). doi: 10.7146/math.scand.a-10841.  Google Scholar

[19]

J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992). doi: 10.7146/math.scand.a-12375.  Google Scholar

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